perturbative aspects of the qcd phase diagram€¦ · perturbative aspects of theqcd phase diagram...

Perturbative aspects of the QCD phase diagram

Urko Reinosa∗

(based on collaborations with J. Serreau, M. Tissier and N. Wschebor)

∗Centre de Physique Théorique, Ecole Polytechnique, CNRS.

GDR-QCD, November 10, 2016, IPNO.

Theoretical approaches to the QCD phase diagram

Non-exhaustive list:

- finite-temperature lattice QCD- Schwinger-Dyson equations- functional renormalization group

⎫⎪⎪⎪⎬⎪⎪⎪⎭non-perturbative

Perturbation theory?

Outline

I. A modified perturbation theory in the infrared?

(Vacuum, Landau gauge).

II. Application to the QCD phase structure.

(Finite T, Landau-DeWitt gauge).

A modified perturbation theoryin the infrared

Perturbation theory: common wisdom

The QCD running coupling decreases at high energies: asymptotic freedom.

As a counterpart, the (perturbative) running coupling increases when theenergy is decreased, and even diverges at a Landau pole known as ΛQCD:

LQCD

E

g2HEL

Yes but ... perturbation theory is based on the Faddeev-Popov procedure whichis at best valid at high energies and should be modified at low energies.

Perturbation theory: (not so) common wisdom

Perturbation theory defined only within a given gauge-fixing (ex: ∂µAaµ = 0).

The gauge-fixing is based on the Faddeev-Popov approach:

A

Equivalent configs AUµ

Field configs satisfyinggauge cond. (∂µAU

µ = 0)

1

⇒ L = 14g2

FaµνFa

µν + ∂µca(Dµc)a + iha∂µAaµ

Perturbation theory: (not so) common wisdom

Perturbation theory defined only within a given gauge-fixing (ex: ∂µAaµ = 0).

The gauge-fixing is based on the Faddeev-Popov approach:

A

Gribov copiesGauge condition

equivalent configs

1

⇒ L = 14g2

FaµνFa

µν + ∂µca(Dµc)a + iha∂µAaµ Valid at best at high energies!

Perturbation theory: (not so) common wisdom

Perturbation theory defined only within a given gauge-fixing (ex: ∂µAaµ = 0).

The gauge-fixing is based on the Faddeev-Popov approach:

A

Gribov copiesGauge condition

equivalent configs

1

⇒ L = 14g2

FaµνFa

µν + ∂µca(Dµc)a + iha∂µAaµ + LGribov ← not known so far

Could LGribov restore the applicability of perturbation theory in the IR?

How to constrain LGribov?

(Landau gauge) lattice simulations [Cucchieri and Mendes; Bogolubsky et al; Dudal et al; . . . ]are crucial: ⋆ free from the Gribov ambiguity;

⋆ provide valuable information to construct models for LGribov.

In particular, they predict:

⋆ a gluon propagator G(p) that behaves like a massive one at p = 0;

⋆ a ghost propagator F(p)/p2 that remains massless.

G(p

)

p (GeV)

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3

F(p

)

p (GeV)

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8

Our model

As the dominant contribution to LGribov, we propose a gluon mass term:

L = 14g2

FaµνFa

µν + ∂µca(Dµc)a + iha∂µAaµ +

12

m2AaµAa

µ

Particular case of the Curci-Ferrari (CF) model. Most appealing features:

⋆ Just one additional parameter (simple modification of the Feynman rules).⋆ Perturbatively renormalizable and ∃ RG trajectories without Landau pole:

Landaupole

No Landau pole

60

40

20

0 0 2 4 6 8 10

g

m

⇒ perturbative calculations may be pushed down to the IR!

Vacuum correlators

Amazing agreement between LO perturbation theory in the CF model andLandau gauge lattice vacuum correlators [with m ≃ 500 MeV for SU(3)]:

G(p

)

p (GeV)

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3

F(p

)

p (GeV)

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8

[Tissier, Wschebor, Phys.Rev. D84 (2011)]

What are the predictions of the model at finite temperature?

QCD phase structure

Back to the QCD phase diagram

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

chiral2nd orderZ(2)

deconfined2nd orderZ(2)

crossover

1st

d

tric

∞

∞

Pure

Order parameter(s): Polyakov loop(s)

` ≡ 13⟨trP eig ∫

1/T0 dτ A0⟩∝ e−βFquark

¯≡ 13⟨tr (P eig ∫

1/T0 dτ A0)

†⟩∝ e−βFantiquark

F = ¥

confined

F < ¥

deconfined

Tc

0

Temperature

Ord

erp

aram

eter

Back to the QCD phase diagram

phys.point

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

1st

2nd orderO(4) ?

chiral2nd orderZ(2)

deconfined2nd orderZ(2)

crossover

1st

d

tric

∞

∞

Pure

Deconfinement transition≡ SSB of center symmetry.

Problem: center symmetry is notmanifest in the Landau gauge.

Underlying symmetry: center symmetry

AUµ = UAµU† − iU∂µU†

U(τ + 1/T, x) = U(τ, x) ei 2π3 k (k = 0,1,2)

Under such a transformation: `→ ei2π3 k`

Broken

Symmetry

Unbroken

Symmetry

Tc

0

Temperature

Ord

erp

aram

eter

Changing to the Landau-DeWitt gauge

We move from the Landau gauge to the Landau-DeWitt gauge:

0 = ∂µAaµ → 0 = (Dµaµ)a { Dab

µ ≡ ∂µδab + f acbAcµ

aaµ ≡ Aa

µ − Aaµ

with A a given background field configuration.

Faddeev-Popov gauge-fixed action + phenomenological LGribov:

LA =1

4g2FaµνFa

µν + (Dµc)a(Dµc)a + iha(Dµaµ)a+12

m2aaµaa

µ

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶Landau gauge + ∂µ → Dµ and Aµ → aµ

Background and symmetries

In practice, one chooses A complying with the symmetries at finite T:

Aµ(τ, x) = Tδµ0 (r3λ3

2+ r8

λ8

2) ∈ Cartan subalgebra

It is also convenient to work with self-consistent backgrounds: 0 = ⟨aµ⟩A.⋆ shown to be the minima of a certain Gibbs potential Γ[A] ≡ ΓA[a = 0].⋆ this functional is center-symmetric:

Γ[AU] = Γ[A], ∀U(τ + 1/T) = ei 2π3 k U(τ)

As in the textbook situation (ex: Ising-model), SSB occurs when the ground stateof the Gibbs potential moves from a symmetric state to a symmetry breaking one.But what is the (center-)symmetric state in the present case?

Weyl chambers and center symmetry

Weyl chambers and center symmetry

Weyl chambers and center symmetry

Weyl chambers and center symmetry

Weyl chambers and center symmetry

Weyl chambers and center symmetry

One-loop result - SU(3)

One-loop result - SU(3)

T−4Γ[A] = 3F(r3, r8,m/T) − F(r3, r8,0) ∼ { −F(r3, r8) if T ≪ m2F(r3, r8) if T ≫ m

One-loop result - SU(3)

We obtain a first order transition in agreement with lattice predictions:

0 2 Π

3

4 Π

3

2 Π

0

[UR, J. Serreau, M. Tissier and N. Wschebor, Phys.Lett. B742 (2015) 61-68]

Summary of one-loop results

order lattice fRG our model at 1-loopSU(2) 2nd 2nd 2ndSU(3) 1st 1st 1stSU(4) 1st 1st 1stSp(2) 1st 1st 1st

Tc (MeV) lattice fRG(∗) our model at 1-loop(∗∗)

SU(2) 295 230 238SU(3) 270 275 185

(∗) L. Fister and J. M. Pawlowski, Phys.Rev. D88 (2013) 045010.(∗∗) UR, J. Serreau, M. Tissier and N. Wschebor, Phys.Lett. B742 (2015) 61-68.

Summary of two-loop results

order lattice fRG our model at 1-loop our model at 2-loopSU(2) 2nd 2nd 2nd 2ndSU(3) 1st 1st 1st 1stSU(4) 1st 1st 1st 1stSp(2) 1st 1st 1st 1st

Tc (MeV) lattice fRG(∗) our model at 1-loop our model at 2-loop(∗∗)

SU(2) 295 230 238 284SU(3) 270 275 185 254

(∗) L. Fister and J. M. Pawlowski, Phys.Rev. D88 (2013) 045010.(∗∗) UR, J. Serreau, M. Tissier and N. Wschebor, Phys.Rev. D93 (2016) 105002.

Adding (fundamental) quarks

We add Nf (heavy) quarks in the fundamental representation:

L = LLdW +LGribov +Nf

∑f=1

{ψf (∂/ − igAa/ ta +Mf + µγ0)ψf }

Center symmetry is explicitly broken by the boundary conditions:

ψ(1/T, x) = −ψ(0, x)

U(1/T, x) = e−i2π/3U(0, x)

⎫⎪⎪⎪⎬⎪⎪⎪⎭⇒ ψ′(1/T, x) = −e−i2π/3ψ′(0, x)

C-symmetry is also broken as soon as µ ≠ 0:r3

r8

µ = 0: Phase transition

µ = 0: Dependence on the quark masses

M > Mc ∶

2.8 3 3.2 3.4 3.6 3.8

V

r3

first order

M = Mc ∶

2.8 3 3.2 3.4 3.6 3.8

V

r3

second order

M < Mc ∶

2.8 3 3.2 3.4 3.6 3.8

V

r3

crossover

µ = 0: Columbia plot

1 0.9 0.8 0.8

0.9

1

Crossover

1st order

1 − e−Msm

1 − e−Mum

µ = 0: Columbia plot

Nf (Mc/Tc)our model (*) (Mc/Tc)lattice (**) (Mc/Tc)matrix (***) (Mc/Tc)SD (****)

1 6.74 7.22 8.04 1.422 7.59 7.91 8.85 1.833 8.07 8.32 9.33 2.04

(∗) UR, J. Serreau and M. Tissier, Phys.Rev. D92 (2015).(∗∗) M. Fromm, J. Langelage, S. Lottini and O. Philipsen, JHEP 1201 (2012) 042.(∗∗∗) K. Kashiwa, R. D. Pisarski and V. V. Skokov, Phys.Rev. D85 (2012) 114029.(∗∗∗∗) C. S. Fischer, J. Luecker and J. M. Pawlowski, Phys.Rev. D91 (2015) 1, 014024.

µ ∈ iR: Phase transitionLow T

µ ∈ iR: Phase transitionHigh T

µ ∈ iR: Roberge-Weiss transition

2

1

0

0.34

0.36 2

1 0

Arg ℓ

T/m

µi/T

µ ∈ iR: Dependence on the quark masses

M > Mc(0) ∶ 0.36

0.348

0 π/3 2π/3µi/T

T/m

M ∈ [Mc(iπ/3),Mc(0)] ∶ 0.36

0.34

0µi/T

π/3 2π/3

T/m

M = Mc(iπ/3) ∶

0.38

0.36

0.34

0.32

µi/Tπ/3 2π/30

T/m

Same structure as in the lattice study of [P. de Forcrand, O. Philipsen, Phys.Rev.Lett. 105 (2010)]

µ ∈ iR: Tricritical scaling

8

7

6 0 0.2 0.4 0.6 0.8 1 1.2

Mc/Tc

(π/3)2 − (µi/Tc)2

McTc= Mtric.

Ttric.+ K [(π

3 )2 + (µT )2]2/5

our model(∗) lattice(∗∗) SD(∗∗∗)

K 1.85 1.55 0.98Mtric.Ttric.

6.15 6.66 0.41

(∗) UR, J. Serreau and M. Tissier, Phys.Rev. D92 (2015).(∗∗) Fromm et.al., JHEP 1201 (2012) 042.(∗∗∗) Fischer et.al., Phys.Rev. D91 (2015) 1, 014024.

µ ∈ R: Complex backgrounds

We find that, for µ ∈ R, the background r8 should be chosen imaginary:

r3

r8

r3

r8

ir8

µ ∈ iR, r8 ∈ R µ ∈ R, r8 ∈ iR

µ ∈ R: Complex backgrounds

The choice r8 = ir8, r8 ∈ R is crucial for the interpretation of ` and ¯ in terms of Fq

and Fq to hold true [UR, J. Serreau and M. Tissier, Phys.Rev. D92 (2015)]:

`(µ) = e−βFq = er8√

3 + 2e−r8

2√

3 cos(r3/2)3

¯(µ) = e−βFq = e−r8√

3 + 2er8

2√

3 cos(r3/2)3

2

1

0

0.36 0.38

T/m

Fq/m

Fq/m

µ ∈ R: Columbia plot

The critical surface moves towards larger quark masses:

0.8

0.9

1

0.8 0.9 1

1 − e−Msm

1 − e−Mum

µ ∈ R: tricritical scaling

The tricritical scaling survives deep in the µ2 > 0 region:

18

14

10

0 40 80 120

Mc/Tc

(π/3)2 + (µ/Tc)2

Conclusions

Simple one-loop calculations in a model for a Gribov completion of theLandau-DeWitt gauge account for qualitative and quantitative featuresof the QCD phase diagram in the heavy quark limit:

⋆ Correct account of the order parameter in the quenched limit;⋆ Critical line of the Columbia plot at µ = 0;⋆ Roberge-Weiss phase diagram and its mass dependence for µ ∈ iR.

The case µ ∈ R requires the use of complex backgrounds.

Certain aspects require the inclusion of two-loop corrections.

TODO:

⋆ Chiral phase transition? Critical end-point in the (T, µ) diagram?⋆ Hadronic bound states? Glueballs?⋆ How to define the physical space?